Orientador: Dr Professora Maria Fernanda de Almeida Cipriano Salvador Marques, FCT-UNL

Outubro, 2015

Tiago João Páscoa Costa

Princípio dos Grandes Desvios para Fluxos

Estocásticos de Fluídos Viscosos

Dissertação para obtenção do Grau de Mestre em Matemática e Aplicações

Orientador: Dr Professora Maria Fernanda de Almeida Cipriano Salvador Marques, FCT-UNL

Membros do júri:

Professora Dra. Ana Bela Cruzeiro Professor Dr. Manuel Esquível

Pr✐♥❝í♣✐♦ ❞♦s ●r❛♥❞❡s ❉❡s✈✐♦s ♣❛r❛ ❋❧✉①♦s ❊st♦❝ást✐❝♦s ❞❡ ❋❧✉í✲ ❞♦s ❱✐s❝♦s♦s

❈♦♣②r✐❣❤t ❝ ❚✐❛❣♦ ❏♦ã♦ Pás❝♦❛ ❈♦st❛✱ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦✲

❣✐❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ◆♦✈❛ ❞❡ ▲✐s❜♦❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❚❛♥t♦ ❛ ❡s❝♦❧❤❛ ❞❡ ✈✐r ♣❛r❛ ♦ ♠❡str❛❞♦ ❞❡ ♠❛t❡♠át✐❝❛ ❡ ❛♣❧✐❝❛çõ❡s ♣❛r❛ ♦ r❛♠♦ ❞❡ ❛♥á❧✐s❡ ♥✉♠ér✐❝❛ ❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ❝♦♠♦ ❛ ❡s❝♦❧❤❛ ❞❡ ❢❛③❡r ❛ t❡s❡ ❞❡ ♠❡str❛❞♦ ♥❡st❡ t❡♠❛ ❢♦r❛♠ ❞❡ ❢❛❝t♦ ✉♠ ❞❡s❛✜♦✳ ❉❡s❛✜♦ ❡st❡✱ q✉❡ ❞✐✜❝✐❧♠❡♥t❡ t❡r✐❛ s✐❞♦ ♣♦ssí✈❡❧ s❡ ♥ã♦ ❢♦ss❡ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❡ss♦❛s q✉❡ ♠❡ ❛❝♦♠♣❛♥❤❛r❛♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❝❡ss♦✳

❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r à Pr♦❢❡ss♦r❛ ❉r✳ ❋❡r♥❛♥❞❛ ❈✐✲ ♣r✐❛♥♦✱ ♥ã♦ só ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❝♦♥st❛♥t❡ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡♠♦♥str❛❞❛✱ ♠❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r s❡♠♣r❡ ❡st❛r ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ♠❡ ❡♥s✐♥❛r tr❛♥s♠✐✲ t✐♥❞♦ ♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠ ✉♠ ✐♥t❡r❡ss❡ ❡ ❡♥t✉s✐❛s♠♦ ú♥✐❝♦s✱ q✉❡ ❧❤❡ sã♦ ❝❛r❛❝t❡ríst✐❝♦s✳

●♦st❛r✐❛ t❛♠❜é♠ ❞❡ ❛❣r❛❞❡❝❡r ❛♦s Pr♦❢❡ss♦r❡s ❉r✳ P❛✉❧❛ ❘♦❞r✐❣✉❡s ❡ ❉r✳ ❋á❜✐♦ ❈❤❛❧✉❜ q✉❡ ♠❡ ♦r✐❡♥t❛r❛♠ ♥✉♠ ♣r♦❥❡t♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ q✉❡ ❞❡s❡♥✈♦❧✈✐ ❡♠ ♣❛r❛❧❡❧♦ ❝♦♠ ❛ t❡s❡✳ ❆❣r❛❞❡ç♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❞❡♠♦♥str❛❞❛✱ ♣❡❧❛ ♦♣♦rt✉✲ ♥✐❞❛❞❡ ❞❡ tr❛❜❛❧❤❛r ♥✉♠❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ✐♥✐❝✐❛❧♠❡♥t❡ ❞❡s❝♦♥❤❡❝✐❛ ♠❛s ♦♥❞❡ ❛❝❛❜❡✐ ♣♦r ❞❡s❝♦❜r✐r ✉♠ ❣r❛♥❞❡ ❢❛s❝í♥✐♦✱ ❡ ♣♦r t♦❞♦ ♦ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡r❛♠ ❢♦rç❛ ♣❛r❛ ❝♦♥t✐♥✉❛r ♦ tr❛❜❛❧❤♦✳

❆♦s ❛♠✐❣♦s✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❥♦r♥❛❞❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦ ❉❛✲ ✈✐❞ ❙♦❛r❡s ♣♦r t♦❞❛s ❛s ❞✐s❝✉ssõ❡s ❡ ✐❞❡✐❛s ♣❛rt✐❧❤❛❞❛s✳

P♦r ú❧t✐♠♦ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ q✉❡r✐❛ t❛♠❜é♠ ❛❣r❛❞❡❝❡r à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r ❡st❛r❡♠ ♣r❡s❡♥t❡s s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳

❘❡s✉♠♦

◆❛ ❧✐t❡r❛t✉r❛ ❡①✐st❡♠ ❞✉❛s ❛❜♦r❞❛❣❡♥s ❛♦ ❡st✉❞♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦s ✢✉í✲ ❞♦s✿ ♣♦r ✉♠ ❧❛❞♦✱ ❛ ❞❡s✐❣♥❛❞❛ ❛♣r♦①✐♠❛çã♦ ❊✉❧❡r✐❛♥❛✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡✲ t❡r♠✐♥❛r ❡ ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝❡rt❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s✱ t❛✐s ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡✱ ♣r❡ssã♦✱ ❡t❝✳ ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦ ✜①♦ x ❞♦ ❡s♣❛ç♦ ❡ ♥✉♠

❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡ t❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❛♣r♦①✐♠❛çã♦ ▲❛❣r❛♥❣❡❛♥❛✱ ♦♥❞❡ ♦

✢✉í❞♦ é ✈✐st♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rtí❝✉❧❛s q✉❡ ♣❛rt❡♠ ❞❡ ♣♦s✐çõ❡s ✐♥✐✲ ❝✐❛✐s ❡ à ♠❡❞✐❞❛ q✉❡ ♦ t❡♠♣♦ ❡✈♦❧✉✐ ❞❡s❝r❡✈❡♠ tr❛❥❡tór✐❛s ♥♦ ♣❧❛♥♦ ♦✉ ♥♦ ❡s♣❛ç♦✳ ◆❡st❛ ♣❡rs♣❡t✐✈❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦ ✢✉í❞♦ é ✈✐st♦ ❝♦♠♦ ✉♠ ✢✉①♦ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦s ♦✉ ❞✐❢❡♦♠♦r✜s♠♦s s♦❜r❡ ❛ r❡❣✐ã♦ ♦❝✉♣❛❞❛ ♣❡❧♦ ✢✉í❞♦✳

❚r❛❜❛❧❤♦s r❡❝❡♥t❡s ✭❝❢✳ ❬✷❪ ✮ r❡✈❡❧❛♠ q✉❡✱ ♣❛r❛ ❝❡rt♦s ✢✉í❞♦s ✈✐s❝♦s♦s ✐♥❝♦♠♣r❡ssí✈❡✐s✱ ❛♣❡s❛r ❞❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s ❊✉❧❡r✐❛♥❛s s❡r❡♠ ❞❡t❡r♠✐♥ís✲ t✐❝❛s✱ ♦ ♠♦✈✐♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s é ✐♥tr✐♥s❡❝❛♠❡♥t❡ ❞❡ ♥❛t✉r❡③❛ ❡st♦❝ást✐❝❛✳ ❆ss✐♠✱ ♥❛ s✉❛ ❞❡s❝r✐çã♦ ▲❛❣r❛♥❣❡❛♥❛ ❞❡✈❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ✢✉①♦s ❡st♦❝ást✐✲ ❝♦s ❞❡✜♥✐❞♦s ❝♦♠♦ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡st♦❝ást✐❝❛s✱ ❝✉❥♦ ❞r✐❢t é ♦ ❝❛♠♣♦ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❞❡✜♥✐❞♦ ❝♦♠♦ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ ❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞✐❢✉sã♦ é ♣r♦♣♦r❝✐♦♥❛❧ à r❛✐③ q✉❛❞r❛❞❛ ❞❛ ✈✐s❝♦s✐❞❛❞❡✳

❯♠ ❞♦s ♣r♦❜❧❡♠❛s ❝❡♥tr❛✐s ❡♠ ♠❡❝â♥✐❝❛ ❞❡ ✢✉í❞♦s é ♦ ❢❡♥ó♠❡♥♦ ❞❛ t✉r✲ ❜✉❧ê♥❝✐❛✱ q✉❡ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦ ♣❛ss❛ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❝♦♠✲ ♣♦rt❛♠❡♥t♦ ❛ss✐♠♣tót✐❝♦ ❞♦s ✢✉í❞♦s ✈✐s❝♦s♦s q✉❛♥❞♦ ❛ ✈✐s❝♦s✐❞❛❞❡ t❡♥❞❡ ♣❛r❛ ③❡r♦✳

◆♦ ❝♦♥t❡①t♦ ❊✉❧❡r✐❛♥♦✱ ♦ ❡st✉❞♦ ❛ss✐♠♣tót✐❝♦ ❞❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s q✉❛♥❞♦ ❛ ✈✐s❝♦s✐❞❛❞❡ t❡♥❞❡ ♣❛r❛ ③❡r♦ é ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦✱ ❛✐♥❞❛ ♥ã♦ r❡s♦❧✈✐❞♦ ❡♠ ❞✐♠❡♥sã♦ três ❡ ❡♠ ❞✐♠❡♥sã♦ ❞♦✐s ♥♦ ❝❛s♦ ❞❡ ❝♦♥❞✐✲ çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t✳ ❊♠ ❞♦♠í♥✐♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ♣❡r✐ó❞✐❝❛s ♦✉ ❞❡ s❧✐♣✱ ❡stá ♣r♦✈❛❞♦ q✉❡ ❛ s♦❧✉çã♦ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s ❝♦♥✈❡r❣❡ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✱ q✉❛♥❞♦ ❛ ✈✐s❝♦s✐❞❛❞❡ t❡♥❞❡ ♣❛r❛ ③❡r♦✳

◆❡st❡ tr❛❜❛❧❤♦ é ❝♦♥s✐❞❡r❛❞❛ ❛ ❛♣r♦①✐♠❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ❡st♦❝ást✐❝❛ ♣❛r❛ ✢✉í❞♦s ✈✐s❝♦s♦s ✐♥❝♦♠♣r❡ssí✈❡✐s✱ ❡ ♣r❡t❡♥❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠ t❡♦r❡♠❛ ❞❡ ❙❝❤✐❧❞❡r ♣❛r❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♠♣tót✐❝♦ ❞❡ ✢✉①♦s ❡st♦❝ást✐❝♦s ❞❡✜♥✐❞♦s ♣❡❧♦ ❝❛♠♣♦ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ q✉❛♥❞♦ ❛ ✈✐s❝♦s✐❞❛❞❡ t❡♥❞❡ ♣❛r❛ ③❡r♦✳

◆♦t❡✲s❡ q✉❡ ❡st❛♠♦s ♣❡r❛♥t❡ ✉♠ ✢✉①♦ ❞❡✜♥✐❞♦ ❛tr❛✈és ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡st♦❝ást✐❝❛ ❡♠ q✉❡ ♦ ❞r✐❢t✱ s❡♥❞♦ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ r✉í❞♦ ❲ ❝♦♥s✐❞❡r❛❞♦ é ✉♠ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ✉♠ ❇r♦✇♥✐❛♥♦ ❝✐❧í♥❞r✐❝♦ ✭❝♦♥str✉í❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠ ♥ú♠❡r♦

✐♥✜♥✐t♦ ❞❡ ♠♦✈✐♠❡♥t♦s ❇r♦✇♥✐❛♥♦s✮✳ ◆❡st❡ ❝♦♥t❡①t♦ ✐rr❡❣✉❧❛r✱ ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ ✢✉①♦ ♥ã♦ ❞❡❝♦rr❡ ❞❡ ♠ét♦❞♦s ❝❧áss✐❝♦s✳ ➱ ✐♠♣♦rt❛♥t❡ r❡❛❧ç❛r q✉❡ ♥❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞♦s ❣r❛♥❞❡s ❞❡s✈✐♦s✱ ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❡♥✈♦❧✈✐❞❛s é ✉♠ r❡q✉✐s✐t♦ ❢✉♥❞❛♠❡♥t❛❧✳

❆ss✐♠✱ ♥♦ ❈❛♣ít✉❧♦ ✸ ✈❛♠♦s ✉s❛r ♦s ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ❬✺❪✱ ❬✶✽❪ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡st♦✲ ❝ást✐❝❛✳

◆♦ ❈❛♣ít✉❧♦ ✹ ❡st❛❜❡❧❡❝❡♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞♦s ❣r❛♥❞❡s ❞❡s✈✐♦s✳ ❆ té❝♥✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ♣r♦✈❛r ♦ ♣r✐♥❝í♣✐♦ ❞♦s ❣r❛♥❞❡s ❞❡s✈✐♦s é ❛ ❛❜♦r❞❛❣❡♠ ❛tr❛✈és ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❘✳ P✳ ❉✉♣✉✐s ❡ ❊❧❧✐s ❬✽❪✱ ❝♦♠ ❜❛s❡ ♥♦ ♣r✐♥❝í♣✐♦ ❞❡ ▲❛♣❧❛❝❡✳ ❊st❡s ♠ét♦❞♦s tê♠✲s❡ r❡✈❡❧❛❞♦ ❛❧t❛♠❡♥t❡ ❡✜❝✐❡♥t❡s ♥♦ ❝❛s♦ ❞❡ ❡q✉❛çõ❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐rr❡❣✉❧❛r❡s✳

◆♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝❛♣ít✉❧♦s ❞❛ t❡s❡ ❝♦❧❡❝✐♦♥❛♠♦s ♦s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s q✉❡ ❥✉❧❣❛♠♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ tó♣✐❝♦ ❡st✉❞❛❞♦ ❡ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡st✉❞♦ ❛ ❞❡s❡♥✈♦❧✈❡r ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ❆s ❞❡♠♦♥str❛çõ❡s ♠❛✐s s✐♠♣❧❡s s❡rã♦ ❛♣r❡s❡♥t❛❞❛s✳

❆❜str❛❝t

■♥ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡r❡ ❛r❡ t✇♦ ♠❛❥♦r ❛♣♣r♦❛❝❤❡s t♦ t❤❡ st✉❞② ♦❢ ✢✉✐❞ ♠♦✈❡✲ ♠❡♥t✿ ✜rst❧②✱ t❤❡ ❞❡s✐❣♥❛t❡❞ ❊✉❧❡r✐❛♥ ❛♣♣r♦❛❝❤✱ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ❞❡t❡r♠✐✲ ♥✐♥❣ ❛♥❞ st✉❞②✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❝❡rt❛✐♥ ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s s✉❝❤ ❛s s♣❡❡❞✱ ♣r❡ss✉r❡✱ ❡t❝✳ ✐♥ ❛ ❝❡rt❛✐♥ ✜①❡❞ ♣♦✐♥tx♦❢ s♣❛❝❡ ❛♥❞ ❛ ❣✐✈❡♥ t✐♠❡t❀ s❡❝♦♥❞❧②✱

t❤❡ ▲❛❣r❛♥❣❡ ❛♣♣r♦❛❝❤✱ ✇❤❡r❡ t❤❡ ✢✉✐❞ ✐s s❡❡♥ ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣❛rt✐❝❧❡s t❤❛t ❧❡❛✈❡ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ❛♥❞ ❛s t✐♠❡ ♣r♦❣r❡ss❡s ❞❡s❝r✐❜❡ tr❛❥❡❝t♦r✐❡s ✐♥ ♣❧❛♥❡ ♦r s♣❛❝❡✳ ■♥ t❤✐s ♣❡rs♣❡❝t✐✈❡ t❤❡ ✢✉✐❞ ♠♦t✐♦♥ ✐s s❡❡♥ ❛s ❛ ✢♦✇ ♦❢ ❤♦♠❡♦♠♦r♣❤✐s♠s ♦r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦♥ t❤❡ r❡❣✐♦♥ ♦❝❝✉♣✐❡❞ ❜② t❤❡ ✢✉✐❞✳

❘❡❝❡♥t ✇♦r❦s ✭s❡❡ ❬✷❪✮ s❤♦✇ t❤❛t✱ ❢♦r ❝❡rt❛✐♥ ✐♠❝♦♠♣r❡ss✐❜❧❡ ✈✐s❝♦✉s ✢✉✐❞s✱ ❞❡s♣✐t❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥❛t✉r❡ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❊✉❧❡r✐❛♥ q✉❛♥t✐t✐❡s✱ t❤❡ ♠♦✲ ✈✐♠❡♥t ♦❢ ♣❛rt✐❝❧❡s ✐s ✐♥❤❡r❡♥t❧② st♦❝❤❛st✐❝ ✐♥ ♥❛t✉r❡✳ ❚❤✉s✱ ✐♥ ✐ts ▲❛❣r❛❣✐❛♥ ❞❡s❝r✐♣t✐♦♥✱ st♦❝❤❛st✐❝ ✢♦✇s ❛s s♦❧✉t✐♦♥s ♦❢ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ❞r✐❢t t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ❞❡✜♥❡❞ ❛s t❤❡ s♦✲ ❧✉t✐♦♥ ♦❢ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥✱ ❛♥❞ t❤❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ✐s ♣r♦♣r♦t✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ r♦♦t ♦❢ t❤❡ ✈✐s❝♦s✐t②✳

❖♥❡ ♦❢ t❤❡ ❝❡♥tr❛❧ ♣r♦❜❧❡♠s ✐♥ ✢✉✐❞ ♠❡❝❤❛♥✐❝s ✐s t❤❡ ♣❤❡♥♦♠❡♥♦♥ ♦❢ t✉r❜✉❧❡♥❝❡✱ ✇❤✐❝❤ ❢r♦♠ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐♥✈♦❧✈❡s t❤❡ ✉♥❞❡rs✲ t❛♥❞✐♥❣ ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ✈✐s❝♦✉s ✢✉✐❞s ✇❤❡♥ t❤❡ ✈✐s❝♦s✐t② t❡♥❞s t♦ ③❡r♦✳

■♥ ❊✉❧❡r✐❛♥ ❝♦♥t❡①t✱ t❤❡ ❛s②♠♣t♦t✐❝ st✉❞② ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ ◆❛✈✐❡r✲ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✇❤❡♥ t❤❡ ✈✐s❝♦s✐t② ❛♣♣r♦❛❝❤❡s ③❡r♦ ✐s ❛ ❝❧❛ss✐❝ ♣r♦❜❧❡♠✱ st✐❧❧ ✉♥s♦❧✈❡❞ ✐♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❞✐♠❡♥s✐♦♥ t✇♦ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❉✐r✐❝❤✲ ❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❖♥ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞♦♠❛✐♥ ✇✐t❤ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ♦r s❧✐♣ ✐s ♣r♦✈❡♥ t❤❛t t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦❧✉t✐♦♥ ❝♦♥✈❡r❣❡s t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✇❤❡♥ t❤❡ ✈✐s❝♦s✐t② t❡♥❞s t♦ ③❡r♦✳

■♥ t❤✐s ✇♦r❦ ✐s ❝♦♥s✐❞❡r❡❞ t❤❡ st♦❝❤❛st✐❝ ▲❛❣r❛♥❣✐❛♥ ❛♣♣r♦❛❝❤ ❢♦r ✐♥❝♦♠✲ ♣r❡ss✐❜❧❡ ✈✐s❝♦✉s ✢✉✐❞s✱ ❛♥❞ ❛✐♠s t♦ ❡st❛❜❧✐s❤ ❛ ❙❝❤✐❧❞❡r✬s t❤❡♦r❡♠ ❢♦r t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ st♦❝❤❛st✐❝ ✢♦✇s ❞❡✜♥❡❞ ❜② t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ✜❡❧❞ ♦❢ ✈❡❧♦❝✐t✐❡s✱ ✇❤❡♥ t❤❡ ✈✐s❝♦s✐t② t❡♥❞s t♦ ③❡r♦✳

◆♦t❡ t❤❛t ✇❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❛ ✢♦✇ ❞❡✜♥❡❞ ❜② t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡ ❞r✐❢t✱ ❜❡✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ◆❛✈✐❡r✲ ❙t♦❦❡s ❡q✉❛t✐♦♥✱ ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ♥♦✐s❡ ❲ ❝♦♥s✐❞❡r❡❞ ✐s ❛ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ♠♦r❡ ♣r❡❝✐s❡❧② ❛ ❝②❧✐♥❞r✐❝❛❧ ❇r♦✇♥✐❛♥✳ ■♥ t❤✐s ✐rr❡❣✉❧❛r ❝♦♥t❡①t✱ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ✢♦✇ ❞♦❡s ♥♦t ❢♦❧❧♦✇ t❤❡ ❝❧❛ss✐❝❛❧ ♠❡t❤♦❞s✳

■t ✐s ♥♦t❡✇♦rt❤② t❤❛t ✐♥ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❧❛r❣❡ ❞❡✈✐❛t✐✲ ♦♥s✱ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈❡❞ ✐s ❛ ❢✉♥❞❛♠❡♥t❛❧ r❡q✉✐r❡♠❡♥t✳

❚❤✉s✱ ✐♥ ❈❤❛♣t❡r ✸ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ♠❡t❤♦❞ ❞❡✈❡❧♦♣❡❞ ❜② ❬✺❪✱ ❬✶✽❪ t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✳

■♥ ❝❤❛♣t❡r ✹ ✇❡ ❡st❛❜❧✐s❤❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s✳ ❚❤❡ t❡❝❤♥✐✲ q✉❡ ✉s❡❞ t♦ ♣r♦✈❡ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ✐s t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ❛♣♣r♦❛❝❤ ❞❡✈❡❧♦♣❡❞ ❜② ❉✉♣✉✐s ❛♥❞ ❊❧❧✐s ❬✽❪✱ ❜❛s❡❞ ♦♥ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ▲❛✲ ♣❧❛❝❡✳ ❚❤❡s❡ ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ❤✐❣❤❧② ❡✣❝✐❡♥t ✐♥ t❤❡ ❝❛s❡ ♦❢ ❡q✉❛t✐♦♥s ✇✐t❤ ✐rr❡❣✉❧❛r ❝♦❡✣❝✐❡♥ts✳

■♥ t❤❡ ✜rst t✇♦ ❝❤❛♣t❡rs ♦❢ t❤❡ t❤❡s✐s ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ t❤❡ ❝❧❛ss✐❝❛❧ r❡s✉❧ts t❤❛t ✇❡ ❝♦♥s✐❞❡r r❡❧❡✈❛♥t t♦ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ t♦♣✐❝ st✉❞✐❡❞ ❛♥❞ ♥❡❝❡ss❛r② ❢♦r t❤❡ st✉❞② ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❚❤❡ s✐♠♣❧❡st ♣r♦♦❢s ❛r❡ ♣r❡s❡♥t❡❞✳

❈♦♥t❡ú❞♦

✶ Pr❡❧✐♠✐♥❛r❡s ✶✶

✶✳✶ ❖♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❡ ❞❡ ❝❧❛ss❡ tr❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ▼❡❞✐❞❛s ●❛✉ss✐❛♥❛s ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸ Pr♦❝❡ss♦s ❊st♦❝ást✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸✳✶ Pr♦❝❡ss♦s ❝♦♠ ✜❧tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ Pr♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺ ■♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✺✳✶ ❉❡✜♥✐çã♦ ❞♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✺✳✷ ❋ór♠✉❧❛ ❞❡ ■tô ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✺✳✸ ❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✺✳✹ ❚❡♦r❡♠❛ ❞❡ ●✐rs❛♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✷ Pr✐♥❝í♣✐♦ ❞❡ ▲❛♣❧❛❝❡ ✸✶

✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ ❋♦r♠✉❧❛çã♦ ❡q✉✐✈❛❧❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸ ❘❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹ ❊♥tr♦♣✐❛ r❡❧❛t✐✈❛ ❡ r❡♣r❡s❡♥t❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❝❡ss♦ ❞❡

❲✐❡♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✸ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ ❋❧✉①♦ ❊st♦❝ást✐❝♦ ✹✸ ✸✳✶ ❋♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✹ Pr✐♥❝í♣✐♦ ❞❡ ❣r❛♥❞❡s ❞❡s✈✐♦s ✻✶

✹✳✶ ❘❛t❡ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✷ ❘❡♣r❡s❡♥t❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✸ ❚✐❣❤t♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✹ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✐❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✺ ❆♣ê♥❞✐❝❡ ✽✷

✶ Pr❡❧✐♠✐♥❛r❡s

✶✳✶ ❖♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❡ ❞❡ ❝❧❛ss❡ tr❛ç♦

❖♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t

❙❡❥❛ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡♣❛rá✈❡❧✱ ❝♦♠ ♥♦r♠❛ |.| = ph., .i ✳ ❈♦♥✲

s✐❞❡r❛♠♦s ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r A : H → H ❡ r❡♣r❡s❡♥t❛♠♦s ♣♦r A∗ _{♦ s❡✉}

❛❞❥✉♥t♦✳

❚❡♦r❡♠❛ ✶✳✶ ❙❡❥❛♠ {ek} ❡ {dk} ❞✉❛s ❜❛s❡s ♦rt♦♥♦r♠❛❞❛s ❞❡ H✱ ❡♥tã♦✿

∞

X

k=1

|Aek|2 =

∞

X

k=1

|Adk|2 ✭✶✮

❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s |Aek|2 ₌P∞

n=1|hAek, dni|2✱ ❡♥tã♦

∞

X

k=1

|Aek|2 ₌

∞

X

k=1

∞

X

n=1

|hAek, dni|2 ₌

∞

X

k=1

∞

X

n=1

|hek, A∗dni|2

= ∞

X

n=1

∞

X

k=1

|hek, A∗dni|2 ₌

∞

X

n=1

|A∗dn|2_.

❆ss✐♠✱ ♦❜t❡♠♦s

∞

X

k=1

|Aek|2 =

∞

X

k=1 |A∗_d

k|2. ✭✷✮

❈♦♠♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r é ✈❡r✐✜❝❛❞❛ ♣❛r❛ q✉❛✐sq✉❡r ❞✉❛s ❜❛s❡s ♦rt♦♥♦r✲ ♠❛❞❛s ❞❡ H✱ t♦♠❛♥❞♦ {ek}={dk} ❞❡❞✉③✐♠♦s q✉❡✿

∞

X

k=1

|Adk_|2 = ∞

X

k=1

|A∗dk_|2. ✭✸✮

P♦rt❛♥t♦ ♦ r❡s✉❧t❛❞♦ ✭✶✮ é ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❡ ✭✷✮ ❡ ✭✸✮✳

❉❡✜♥✐çã♦ ✶✳✷ ❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r A : H → H ❞✐③✲s❡ ✉♠ ♦♣❡r❛❞♦r ❞❡

❍✐❧❜❡rt✲❙❝❤♠✐❞t s❡ ♣❛r❛ ❛❧❣✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❞❛{ek}❞❡H ✱ P∞k=1|Aek|2 <

∞✳ ❆ ♥♦r♠❛ ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t é ❞❡✜♥✐❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿

||A_||HS =

∞

X

k=1 |Aek|2

!1/2

.

◆♦t❛✿ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✶ ❛ ♥♦r♠❛ ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❞❡ ✉♠ ♦♣❡r❛❞♦rA❡stá

❜❡♠ ❞❡✜♥✐❞❛✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❜❛s❡✳

❚❡♦r❡♠❛ ✶✳✸ ❙❡❥❛♠ ❆ ❡ ❇ ♦♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❡♥tã♦✱ ❛s s❡❣✉✐♥✲ t❡s ❛✜r♠❛çõ❡s ✈❡r✐✜❝❛♠✲s❡✿

(i) _||A∗_||

HS =||A||HS❀

(ii) ||αA||HS =|α|||A||HS✱ α∈R❀

(iii) _||A+B_||HS ≤ ||A||HS +||B||HS❀

(iv) ||A|| ≤ ||A||HS✱ ♦♥❞❡ ||A||= sup x6=0

|Ax| |x_| .

❉❡♠♦♥str❛çã♦✿ ❆ ♣r♦♣r✐❡❞❛❞❡ (i) ❞❡❞✉③✲s❡ ❞✐r❡t❛♠❡♥t❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ❡ (ii) r❡s✉❧t❛ tr✐✈✐❛❧♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ♥♦r♠❛✳

❆ ♣r♦♣♦s✐çã♦ ✭✐✐✐✮ é ♦❜t✐❞❛ ❞♦ ❢❛❝t♦ ❞❡ |(A+B)x| ≤ |Ax|+|Bx| ❡ ❞❛ ❞❡s✐✲

❣✉❛❧❞❛❞❡ ❞❡ ▼✐♥❦♦✇s❦✐✿

∞

X

k=1

|αk+βk|212 _≤

∞

X

k=1

|αk|212 ₊

∞

X

k=1

|βk|212_.

P❛r❛ ❞❡♠♦♥str❛r (iv)♦❜s❡r✈❛♠♦s q✉❡

|Ax|2 ₌

∞

X

k=1

|hAx, eki|2 ₌

∞

X

k=1

|hx, A∗eki|2 _≤

∞

X

k=1

|x|2_|A∗_ek|2

=|x|2

∞

X

k=1

|A∗ek|2 _=|x|2_||A∗_||2

2 =|x|2||A||22.

P♦rt❛♥t♦✱ |Ax| ≤ |x|||A||HS✳

❱❛♠♦s ❡♥tã♦ ❞❡♥♦t❛r L(2)(H) ❝♦♠♦ ♦ ❡s♣❛ç♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲

❙❝❤♠✐❞t ❞❡ H ❡ ♣♦r L(H) ♦ ❡s♣❛ç♦ ❞♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❝♦♥tí♥✉♦s ❞❡ H✳

◆♦t❛✿ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✸ (iv) t❡♠♦s L(2)(H) ⊂ L(H)✳ ❙❡ H t❡♠ ❞✐♠❡♥sã♦

✜♥✐t❛✱ ❡♥tã♦ L(2)(H) = L(H)✳ ❙❡ H t❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ♥ã♦ é ✈❡r✐✜❝❛❞❛ ❛

✐❣✉❛❧❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✱ ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡I ❞❡H ♣❡rt❡♥❝❡ ❛L(H)♠❛s

♥ã♦ ♣❡rt❡♥❝❡ ❛ L(2)(H)✳

❉❡✜♥✐çã♦ ✶✳✹ ❙❡❥❛♠ A, B ∈ L(2)(H)✳ ❉❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡

❍✐❧❜❡rt✲❙❝❤♠✐❞t hA, B_iHS ♣♦r

hA, BiHS =

∞

X

k=1

hAek, Beki

❖♥❞❡ {ek_} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❞❛ ❞❡ H✳

◆♦t❛✿ ➱ ✐♠❡❞✐❛t♦ q✉❡ ❛ sér✐❡ ❛♥t❡r✐♦r ❝♦♥✈❡r❣❡✱ ♣♦✐s✱ hAek, Beki ≤

|Aek_|2 _+|Bek|2_{✳ ❯s❛♥❞♦ ❛r❣✉♠❡♥t♦s s❡♠❡❧❤❛♥t❡s ❛♦s ❞♦ ❚❡♦r❡♠❛ ✶✳✶✱ t❡✲}

♠♦s t❛♠❜é♠ q✉❡ hA, BiHS ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳

❚❡♦r❡♠❛ ✶✳✺ L(2)(H)❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦h., .iHS é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

❉❡♠♦♥str❛çã♦✿ P❡❧❛s ❛❧í♥❡❛s (ii) ❡ (iii) ❞♦ ❚❡♦r❡♠❛ ✶✳✸ ✈❡r✐✜❝❛♠♦s q✉❡

L(2)(H)é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❚❡♠♦s t❛♠❜é♠ q✉❡hA, Ai=||A||2HS✳ P♦rt❛♥t♦

só t❡♠♦s q✉❡ ❞❡♠♦♥str❛r q✉❡ L(2)(H) é ❝♦♠♣❧❡t♦✳

❙❡❥❛ {An} ✉♠❛ s✉❝❡ssã♦ ❞❡ ❈❛✉❝❤② ❡♠ L(2)(H) ❡♥tã♦✱ ♣❡❧❛ ❛❧í♥❡❛ (iv) ❞♦

❚❡♦r❡♠❛ ✶✳✸ ❛ s✉❝❡ssã♦ {Ak} é ✉♠❛ s✉❝❡ssã♦ ❞❡ ❈❛✉❝❤② ❡♠ L(H)✳ ❈♦♠♦

L(H)❝♦♠ ❛ ♥♦r♠❛ ❞♦ ♦♣❡r❛❞♦r é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡①✐st❡A_∈L(H)t❛❧

q✉❡limn→∞||An−A||= 0✳ ❱❛♠♦s ✈❡r q✉❡ A∈L(2)(H)❡ q✉❡ limn→∞||An−

A||HS = 0✳ ❋✐①❡♠♦s ǫ > 0✱ ❡♥tã♦ ♣❛r❛ m, n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s ||An− Am_||HS < ǫ✳ ▲♦❣♦

s

X

k=1

|(An−Am)ek|2 _{≤ ||An−Am||}2

HS < ǫ2

♣❛r❛ q✉❛❧q✉❡rs❡m, ns✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✳ ❈♦♠♦limn→∞||An−A||= 0

❢❛③❡♠♦s m t❡♥❞❡r ❛ ✐♥✜♥✐t♦ ♦❜t❡♥❞♦ s

X

k=1

|(An−A)ek|2 _≤ǫ2

♣❛r❛ q✉❛❧q✉❡r s✱ ❡ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❋❛③❡♥❞♦ ❛❣♦r❛ s → ∞ t❡♠♦s

q✉❡

∞

X

k=1

|(A−An)ek|2 _≤ǫ2 _<∞.

P♦rt❛♥t♦✱ A ₋An _∈ L(2)(H)✱ ❧♦❣♦ A = An+ (A−An) ∈ L(2)(H) ❡ ❝♦♠♦ ||A −An||2 ≤ ǫ✱ ♣❛r❛ ♥ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ t❡♠♦s q✉❡ limn→∞||An−

A_||HS = 0.

❈❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s tr❛ç♦

❱❛♠♦s ❛❣♦r❛ ✐♥tr♦❞✉③✐r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡✜♥✐r ❛ ❝❧❛ss❡ ❞♦s ♦♣❡r❛❞♦r❡s tr❛ç♦✳

❉❡✜♥✐çã♦ ✶✳✻ ❯♠ ♦♣❡r❛❞♦r A:H _→H ❞✐③✲s❡ ❝♦♠♣❛❝t♦ s❡ ❛ ❝❛❞❛ ❝♦♥❥✉♥t♦

❧✐♠✐t❛❞♦ ❞❡ H ❢❛③ ❝♦rr❡s♣♦♥❞❡r ✉♠ ❝♦♥❥✉♥t♦ ❝✉❥♦ ❢❡❝❤♦ é ❝♦♠♣❛❝t♦✳

❚❡♦r❡♠❛ ✶✳✼ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦✱ ❡♥tã♦ A é ❧✐♠✐t❛❞♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛A✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦✱ ❞❡♥♦t❛♠♦s ♣♦rB(0,1)❛ ❜♦❧❛

❡♠ H ❞❡ ❝❡♥tr♦ 0 ❡ r❛✐♦1✳ ❚❡♠♦s✿

||A_||= sup_{|Au_|: u_∈H _{∧ |}u_{| ≤}1_}= sup_{|Au_|: u_∈B(0,1)_}.

❈♦♠♦B(0,1)é ❧✐♠✐t❛❞♦ ❡Aé ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦✱ ♦ ❢❡❝❤♦ ❞❡ A(B(0,1))

é ❝♦♠♣❛❝t♦ ❡ ♣♦rt❛♥t♦ ❧✐♠✐t❛❞♦✳ ▲♦❣♦ ∃k _∈ R _{: ∀u ∈ B(0,1), |Au| ≤ k}✳

P♦rt❛♥t♦ ||A_{|| ≤}k✳

❉❡✜♥✐çã♦ ✶✳✽ ❯♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦ A ❞❡ A: H _→H ❞✐③✲s❡ ♦♣❡r❛❞♦r ❞❡

❝❧❛ss❡ tr❛ç♦ s❡ P∞

n=1λn <∞✱ ♦♥❞❡ λn sã♦ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡ (A∗A)

1 2✳

✶✳✷ ▼❡❞✐❞❛s ●❛✉ss✐❛♥❛s ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❜❡rt

❉❛❞♦ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✱ r❡♣r❡s❡♥t❛♠♦s ♣♦rB_(H)❛ _σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳ ❯♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ µ ❞❡✜♥✐❞❛ s♦❜r❡

♦ ❡s♣❛ç♦ ♠❡♥s✉rá✈❡❧ (H, B_(H)) é ❞❡♥♦♠✐♥❛❞❛ ♠❡❞✐❞❛ ●❛✉ss✐❛♥❛ s❡ ♣❛r❛

h∈H ❡①✐st✐r❡♠ n∈R1 ❡ _q≥0 t❛✐s q✉❡✱

µ({x∈H;hh, xi ∈A}) =N _{(n, q)(A), ∀A∈}B₍R1₎

♦♥❞❡ N _{(n, q)(·)} ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ●❛✉ss✐❛♥❛ s♦❜r❡ ₍R1_, B₍R1₎₎ ❞❡ ♠é❞✐❛ _n

❡ ✈❛r✐â♥❝✐❛ q:

N _{(n, q)(A) = √}1

2πq

Z

A

❡−(x−n)2 2q dx.

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ µ ❢♦r ●❛✉ss✐❛♥❛✱ ♦s ❢✉♥❝✐♦♥❛✐s

H _→R1_{, h→}

Z

Hh

h, x_iµ(dx),

H_×H _→R1_{, (h1, h2)→}

Z

Hh

h1, xihh2, xiµ(dx),

❡stã♦ ❜❡♠ ❞❡✜♥✐❞♦s✳ ❱❛♠♦s ❛❣♦r❛ ✈❡r q✉❡ ❡st❡s ❢✉♥❝✐♦♥❛✐s sã♦ ❝♦♥tí♥✉♦s✱ ♣❛r❛ ✐ss♦ ✈❛♠♦s ✐♥tr♦❞✉③✐r ♦ s❡❣✉✐♥t❡ ❧❡♠❛ s♦❜r❡ ♠❡❞✐❞❛s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ▲❡♠❛ ✶✳✾ ❙❡❥❛ ν ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♠ (H, B_(H))✳ ❱❛♠♦s ❛s✲ s✉♠✐r q✉❡ ♣❛r❛ ❛❧❣✉♠ k _∈N

Z

H|h

z, x_i|kν(dx)<+_∞, _∀z _∈H.

❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c >0 t❛❧ q✉❡

Z

H

hh1, xi...hhk, xiν(dx)

≤c|h1|...|hk|, h1...hk∈H.

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛Un ♣❛r❛ n_∈N ♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r

Un =

z _∈H :

Z

H|h

z, x_i|kν(dx)_≤n

.

P♦r ❤✐♣ót❡s❡ H = S∞_n=1Un✳ ❈♦♠♦ H é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡ Un

sã♦ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s✱ ♣❡❧♦ ❛r❣✉♠❡♥t♦ ❞❡ ❝❛t❡❣♦r✐❛ ❞❡ ❇❛✐r❡✱ ❡①✐st❡n0 ∈N✱

z0 ∈Un0 ❡ r0 >0 t❛❧ q✉❡ B(z0, r0)⊂Un0✳ ▲♦❣♦

Z

H|h

z0+y, xi|kν(dx)≤n0, ∀y∈B(0, r0).

▼❛s ♣❛r❛ q✉❛❧q✉❡r y∈B(0, r0)✱ t❡♠♦s

Z

H|h

y, x_i|k_ν(dx)≤2k

Z

H|h

z0+y, xi|kν(dx) + 2k

Z

H|h

z0, xi|kν(dx)≤2k+1n0.

❆ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡r z ∈ H ❞✐❢❡r❡♥t❡ ❞❡ 0 ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡

❛♥t❡r✐♦r ❛ y=r0_|Zz_| ♦❜t❡♥❞♦

Z

H|h

z, x_i|kν(dx)_≤2k+1n0|z|kr−0k.

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡

|ξ1ξ2...ξk| ≤ |ξ1|k+|ξ2|k+...+|ξk|k ∀(ξ1, ξ2, ..., ξk)∈Rk,

❝♦♥st❛t❛♠♦s q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦

Hk _→R1_{, (h}

1, .., hk)→

Z

Hh

h1, xi...hhk, xiν(dx)

é ❝♦♥tí♥✉❛✳

❉❡❝♦rr❡ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r q✉❡ s❡µé ✉♠❛ ♠❡❞✐❞❛ ●❛✉ss✐❛♥❛✱ ❡♥tã♦ ❡①✐st❡

✉♠ ❡❧❡♠❡♥t♦ m _∈H ❡ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ◗✱ t❛❧ q✉❡

Z

Hh

h, x_iµ(dx) = _hm, h_i, _∀h_∈H, ✭✹✮

Z

Hh

h1, x−mihh2, x−miµ(dx) =hQh1, h2i, ∀h1, h2 ∈H. ✭✺✮

❆♦ ✈❡t♦r m ❝❤❛♠❛♠♦s ♠é❞✐❛ ❡ ❛ Q ♦ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❞❡ µ✳ ❖

♦♣❡r❛❞♦r Q é s✐♠étr✐❝♦ ❡ ❝♦♠♦

hQh, h_i=

Z

Hh

h, x₋m_i2µ(dx)_≥0, h_∈H,

t❛♠❜é♠ é ♥ã♦ ♥❡❣❛t✐✈♦✳ ❘❡s✉❧t❛ ❞❡ (4) ❡ (5) q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❝❛r❛❝t❡ríst✐❝♦

❞❡ ✉♠❛ ♠❡❞✐❞❛ ●❛✉ss✐❛♥❛ µ ❝♦♠ ♠é❞✐❛ m ❡ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛ Q✱

N(m, Q)✱ é ❞❛❞♦ ♣♦r

ˆ

µ(λ) =

Z

H

eihλ,xiµ(dx) = eihλ,mi−12hQλ,λi.

P♦rt❛♥t♦ µˆ é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ ♣♦r m ❡ Q✳ ❆ ♠❡❞✐❞❛ ●❛✉ss✐❛♥❛ ❞❡

♠é❞✐❛ m ❡ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛ Qs❡rá ❞❡♥♦t❛❞❛ ♣♦r N _{(m, Q).}

✶✳✸ Pr♦❝❡ss♦s ❊st♦❝ást✐❝♦s

✶✳✸✳✶ Pr♦❝❡ss♦s ❝♦♠ ✜❧tr❛çã♦

❱❛♠♦s ❛ss✉♠✐r q✉❡I = [0, T] ❡ q✉❡ ♦ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡(Ω,F_,P₎❡stá

❡q✉✐♣❛❞♦ ❝♦♠ ❛ ❢❛♠í❧✐❛ ❝r❡s❝❡♥t❡ ❞❡ σ✲á❧❣❡❜r❛s {F_{t}, t ∈ I}✱ ❛ q✉❛❧ ❝❤❛♠❛✲ ♠♦s ✜❧tr❛çã♦✳ ■r❡♠♦s ❞❡s✐❣♥❛r ♣♦r F_t+ ❛ ✐♥t❡rs❡❝çã♦ ❞❡ t♦❞❛s ❛s F_s ♦♥❞❡

s < t✳ ❯♠❛ ✜❧tr❛çã♦ ❞✐③✲s❡ ♥♦r♠❛❧ s❡✿

(i) F₀ ❝♦♥té♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s _A∈F t❛❧ q✉❡ P_{(A) = 0}❀

(ii) F_{t+ =}F_t ♣❛r❛ q✉❛❧q✉❡r _t∈T✳

❯♠ ♣r♦❝❡ss♦ X ❞✐③✲s❡ ❛❞❛♣t❛❞♦ s❡ ♣❛r❛ q✉❛❧q✉❡r t _∈ I ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tó✲

r✐❛ X(t)é F_t✲♠❡♥s✉rá✈❡❧✳

X ❞✐③✲s❡ ♣r♦❣r❡ss✐✈❛♠❡♥t❡ ♠❡♥s✉rá✈❡❧ s❡ ♣❛r❛ ❝❛❞❛ t∈[0, T] ❛ ❛♣❧✐❝❛çã♦

[0, t]_×Ω_→E, (s, ω)_→X(s, ω)

é B_{([0, t])×}F_t✲♠❡♥s✉rá✈❡❧✳ ❘❡♣r❡s❡♥t❛♠♦s ♣♦rF_∞ ❛ _σ✲á❧❣❡❜r❛ ❞❡ s✉❜❝♦♥✲ ❥✉♥t♦s ❞❡ [0,_∞)_×Ω✱ ❣❡r❛❞❛ ♣❡❧♦s ❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛✿

(s, t]_×F, 0_≤s < t <_∞, F _∈F_s ❡ _{{0} ×F, F ∈}F_.

❊st❛ σ✲á❧❣❡❜r❛ ❞✐③✲s❡ σ✲á❧❣❡❜r❛ ♣r❡✈✐sí✈❡❧ ❡ ♦s s❡✉s ❡❧❡♠❡♥t♦s ❞✐③❡♠✲s❡ ❝♦♥✲

❥✉♥t♦s ♣r❡✈✐sí✈❡✐s✳ ❆ r❡str✐çã♦ ❞❡ F_∞ ❛ _{[0, T]×Ω} ✈❛✐ s❡r ❞❡♥♦♠✐♥❛❞❛ ♣♦r F_T✳

❯♠❛ ❛♣❧✐❝❛çã♦ ♠❡♥s✉rá✈❡❧ ❞❡✜♥✐❞❛ ♥♦ ❡s♣❛ç♦([0, T]×Ω,F_T)❝♦♠ ✈❛❧♦r❡s ❡♠

(E,B_(E)) ❞❡s✐❣♥❛✲s❡ ✉♠ ♣r♦❝❡ss♦ ♣r❡✈✐sí✈❡❧✳ ❯♠ ♣r♦❝❡ss♦ ♣r❡✈✐sí✈❡❧ é ♥❡✲ ❝❡ss❛r✐❛♠❡♥t❡ ❛❞❛♣t❛❞♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ r❡❢❡r✐♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ ✐♥❞✐❝❛ ❡♠ q✉❡ ❝♦♥❞✐çõ❡s ✉♠ ♣r♦❝❡ss♦ ❛❞❛♣t❛❞♦ é ♣r❡✈✐sí✈❡❧ ✭❝❢✳ ❬✻❪✮✳

❚❡♦r❡♠❛ ✶✳✶✵

❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿

(i) ❯♠ ♣r♦❝❡ss♦ ❛❞❛♣t❛❞♦ Φ ❝♦♠ ✈❛❧♦r❡s ❡♠ L(U, H) t❛❧ q✉❡✱ ♣❛r❛ u ∈ U

❡ h _∈ H ❛r❜✐trár✐♦s ♦ ♣r♦❝❡ss♦ _hΦ(t)u, h_i, t _≥0 t❡♠ tr❛❥❡tór✐❛s ❝♦♥tí♥✉❛s à

❡sq✉❡r❞❛✱ é ♣r❡✈✐sí✈❡❧✳

(ii) ❙❡❥❛ Φ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝❛st✐❝❛♠❡♥t❡ ❝♦♥tí♥✉♦ ❡ ❛❞❛♣t❛❞♦ ♥♦ ✐♥t❡r✈❛❧♦ [0, T]✳ ❊♥tã♦ ♦ ♣r♦❝❡ss♦ Φ t❡♠ ✉♠❛ ✈❡rsã♦ ♣r❡✈✐sí✈❡❧ ❡♠ [0, T]✳

✶✳✹ Pr♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛

❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛ U ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

h·,·i✳ ❯♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ W(t), t ≥ 0, ❝♦♠ ✈❛❧♦r❡s ❡♠ U ❞✐③✲s❡ ✉♠ Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r s❡✿

✭✐✮ W(0) = 0,

✭✐✐✮ W t❡♠ tr❛❥❡tór✐❛s ❝♦♥tí♥✉❛s✱

✭✐✐✐✮ W t❡♠ ✐♥❝r❡♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱

✭✐✈✮ ❆ ❧❡✐ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛W(t)−W(s) é N _{(0,(t−s)Q), t≥s ≥0,} ♦♥❞❡ ◗ é ✉♠ ♦♣❡r❛❞♦r tr❛ç♦ ♥ã♦ ♥❡❣❛t✐✈♦ ❡♠ U✱ ✐st♦ é✱ ❡①✐st❡ ✉♠❛ ❜❛s❡

♦rt♦♥♦r♠❛❞❛ ❝♦♠♣❧❡t❛ {ek_} ❡♠ U ❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦

♥❡❣❛t✐✈♦s {λk_} t❛❧ q✉❡

X

k∈N

||Qek_||U =

X

k∈N

||λkek_||U <∞.

❊st❡ ♦♣❡r❛❞♦r Q é ❞❡♥♦♠✐♥❛❞♦ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛✳

❚❡♦r❡♠❛ ✶✳✶✷ ❙❡❥❛ W(t) ✉♠Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r✳ ❊♥tã♦ ✈❡r✐✜❝❛♠✲s❡ ❛s

s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

(i) ❲ é ✉♠ ♣r♦❝❡ss♦ ●❛✉ss✐❛♥♦ ❡♠ U ❡

E_{(W(t)) = 0, Cov(W(t)) = tQ, t≥0.}

(ii) P❛r❛ t >0 ❛r❜✐trár✐♦✱ ❲ ❛❞♠✐t❡ ❛ ❡①♣❛♥sã♦

W(t) =X

n∈N

p

λjβj(t)ej, ✭✻✮

♦♥❞❡

βj(t) =

1

p

λj

hW(t), eji, j ∈N

sã♦ ♠♦✈✐♠❡♥t♦s ❇r♦✇♥✐❛♥♦s ❝♦♠ ✈❛❧♦r❡s r❡❛✐s ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ (Ω,F_,P₎✱ ❡ ❛ sér✐❡ ✭✶✮ é ❝♦♥✈❡r❣❡♥t❡✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ 0 < t1 < ... < tn ❡ s❡❥❛ u1, ..., un ∈ U✳ ❈♦♥s✐❞❡r❡♠♦s ❛

✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ Z ❞❡✜♥✐❞❛ ♣♦r

Z =

n

X

k=1

hW(tk), uki=

n

X

k=1

hW(t1), uki+

∞

X

k=2

hW(t2)−W(t1), uki

+...+hW(tn)−W(tn₋1), uni.

❈♦♠♦ W t❡♠ ✐♥❝r❡♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ Z é ●❛✉ss✐❛♥❛ ♣❛r❛ q✉❛❧q✉❡r ❡s✲

❝♦❧❤❛ ❞❡ u1, ..., un ❡ ♦❜t❡♠♦s (i)✳

❱❛♠♦s ❛❣♦r❛ ❞❡♠♦♥str❛r (ii)✳ ❙❡❥❛ t > s >0✱ ❡♥tã♦

E_{(βi(t)βj(s)) =p}1

λiλjE(hW(t), eiihW(s), eji)

=p1

λiλj[E(hW(t)−W(s), eiihW(s), eji)

+E_{(hW(s), eiihW(s), eji)]}

=p1

λiλjshQei, eji=sδij.

P♦rt❛♥t♦✱ t❡♠♦s ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡ βi, i ∈N✳ P❛r❛ ♣r♦✈❛r ❛ r❡♣r❡s❡♥t❛çã♦

(3) é s✉✜❝✐❡♥t❡ ♦❜s❡r✈❛r q✉❡ ♣❛r❛m ≥n≥1

E

m

X

j=n

p

λjβj(t)ej

2

=t m

X

j=n λj;

❡ r❡❧❡♠❜r❛r q✉❡ ♣♦r ❞❡✜♥✐çã♦ P∞

j=1λ <∞.

❱❛♠♦s ❛❣♦r❛ ❣❡♥❡r❛❧✐③❛r ❛ ❞❡✜♥✐çã♦ ♣❛r❛ ♣r♦❝❡ss♦s ❞❡ ❲✐❡♥❡r ♥✉♠ ❡s✲ ♣❛ç♦ ❞❡ ❍✐❧❜❡rt U✱ ♦♥❞❡ ♦ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛ Q ♥ã♦ é ❞❡ ✉♠ ♦♣❡r❛❞♦r

tr❛ç♦✳

❙❡❥❛ W(t) ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt U ❡ Q ♦ s❡✉

♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛a _∈U ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ♣r♦❝❡ss♦

❞❡ ❲✐❡♥❡r ❝♦♠ ✈❛❧♦r❡s r❡❛✐s Wa(t), t_≥0, ♣♦r✿

Wa(t) = hW(t), ai, t≥0.

❆ tr❛♥s❢♦r♠❛çã♦ a_→Wa é ❧✐♥❡❛r ❞❡U ♣❛r❛ ♦ ❡s♣❛ç♦ ❞♦s ♣r♦❝❡ss♦s ❡st♦❝ás✲

t✐❝♦s✳ ➱ ❛✐♥❞❛ ❝♦♥tí♥✉❛ ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦✿

∀t≤0,{an} ⊂U, lim

n→∞an =a ⇒nlim→∞

E_|Wa(t)−Wa

n(t)|

2 _{= 0. ✭✼✮}

◗✉❛❧q✉❡r tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r a→Wa s❛t✐s❢❛③❡♥❞♦ (4) é ❝❤❛♠❛❞❛ ♣r♦❝❡ss♦

❞❡ ❲✐❡♥❡r ❣❡♥❡r❛❧✐③❛❞♦✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r K(a, b)✱ a, b∈U

❡ t > s >0t❛❧ q✉❡ ✿

E_{[hW(t), aihW(s), bi] =}E_{[(hW(t), ai − hW(s), ai)hW(s), bi]}

+E_{[hW(s), aihW(s), bi]}

=sE_{[hW(1), aihW(1), bi] =sK(a, b).}

❆ ❝♦♥❞✐çã♦(4)✐♠♣❧✐❝❛ q✉❡Ké ❜✐❧✐♥❡❛r ❝♦♥tí♥✉❛ ❡♠U✱ ❧♦❣♦ ❡①✐st❡Q∈L(U)✱

t❛❧ q✉❡✿

E_{[Wa(t)Wb(s)] = shQa, bi, t > s≥0, a, b∈U.} ✭✽✮

❖ ♦♣❡r❛❞♦rQ❝❤❛♠❛✲s❡ ❝♦✈❛r✐â♥❝✐❛ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❣❡♥❡r❛❧✐③❛❞♦a→

Wa✳ ❊st❡ ♦♣❡r❛❞♦r é ❛✉t♦ ❛❞❥✉♥t♦ ❡ ❞❡✜♥✐❞♦ ♣♦s✐t✐✈♦✳

❉❛❞♦ ✉♠ ♦♣❡r❛❞♦rQ✱ é r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝✐❧ ❝♦♥str✉✐r ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r

❣❡♥❡r❛❧✐③❛❞♦ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ♠❡♥❝✐♦♥❛❞❛s✳ ❉❡ ❢❛❝t♦✱ s❡❥❛{ej}✉♠❛

❜❛s❡ ♦rt♦♥♦r♠❛❞❛ ❝♦♠♣❧❡t❛ ❞❡U✱_{βj_}✉♠❛ s✉❝❡ssã♦ ❞❡ ♣r♦❝❡ss♦s ❞❡ ❲✐❡♥❡r

❝♦♠ ✈❛❧♦r❡s r❡❛✐s✱ ❡ U0 =Q1/2(U)✱ ❞❡✜♥❡✲s❡✿

Wa(t) =X

j∈N

hQ1/2ej, aiβj, t≥0, a∈U.

❈♦♠♦ _X

j∈N

|hQ1/2ej, a_i|2 =_|Q1/2a_|2 <_∞

♣❛r❛ ❝❛❞❛ a ∈ U✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈❡rsã♦ ❞❡ Wa q✉❡ é ✉♠ ♣r♦❝❡ss♦ ❞❡

❲✐❡♥❡r✳

❚❡♠♦s ❛✐♥❞❛ q✉❡

E_{[hW(t), aihW(s), bi] =s}X

j∈N

hQ1/2ej, aihQ1/2ej, bi=shQa, bi.

❚❡♦r❡♠❛ ✶✳✶✸ ❙❡❥❛ U1 ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ t❛❧ q✉❡ U0 = Q1/2(U) ❡stá

✐♥s❡r✐❞♦ ❡♠ U1 ♣♦r ✉♠❛ ✐♥❝❧✉sã♦ ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t J✳ ❊♥tã♦

W(t) =X

n∈N

Q1/2ejβj(t), t_≥0, ✭✾✮

❞❡✜♥❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❝♦♠ ✈❛❧♦r❡s ❡♠ U1✳ ❆❧é♠ ❞✐ss♦✱ s❡ Q1 ❢♦r ❛

❝♦✈❛r✐â♥❝✐❛ ❞❡ W✱ ❡♥tã♦ Q1₁/2(U1) ❡ Q1/2(U) sã♦ ✐❣✉❛✐s✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ gj = Q1/2ej✱ j ∈ N✱ ❡♥tã♦ {gj} ❢♦r♠❛ ✉♠❛ ❜❛s❡ ♦rt♦✲

♥♦r♠❛❞❛ ❝♦♠♣❧❡t❛ ❡♠ U0✱ ❡ ♣♦rt❛♥t♦

||JQ1/2||H.S =

X

n∈N

|Jgj|2

U1 <∞.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡ (6) ❞❡✜♥❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❡♠ U1✳ P❛r❛a, b∈U1

t❡♠♦s✿

hQa, biU1 =E[hW(1), aiU1hW(1), biU1] =

X

n∈N

hJgj, aiU1hJgj, biU1

=X

n∈N

hgj, J∗aiU0hgj, J∗biU0 =hJ∗a, J∗biU0 =hJJ∗a, biU1,

♦ q✉❡ ✐♠♣❧✐❝❛ JJ∗=Q1✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

|Q1₁/2a_|2_U1 =_hJ∗a, J∗a_iU1 =|J∗a|

2

U0, a∈U1.

❊♥tã♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❆ ✭✈❡r ❆♣ê♥❞✐❝❡✮ ❛♣❧✐❝❛❞❛ ❛ Q1/2 _{: U}

1 → U1 ❡

J :U0 →U1 t❡♠♦s Q11/2(U1) =J(U0) =U0 ❡ |Q−11/2u|U1 =|u|U0✱ t❡r♠✐♥❛♥❞♦

❛ss✐♠ ❛ ❞❡♠♦♥str❛çã♦✳

❊♥tã♦✱ ❛❞♠✐t✐♥❞♦ ✉♠ ❛❜✉s♦ ❞❡ ❧✐♥❣✉❛❣❡♠✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ✉♠ ♣r♦✲ ❝❡ss♦ ❞❡ ❲✐❡♥❡r ❣❡♥❡r❛❧✐③❛❞♦ ❡♠ U é ✉♠Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ♥✉♠ ❡s♣❛ç♦

❞❡ ❍✐❧❜❡rt ♠❛✐♦r U1✳

◗✉❛♥❞♦ ♦ ♦♣❡r❛❞♦r Qé ❛ ✐❞❡♥t✐❞❛❞❡✱ ❞✐③✲s❡ q✉❡ ❡st❡ ♣r♦❝❡ss♦ é ✉♠ ♣r♦❝❡ss♦

❞❡ ❲✐❡♥❡r ❝✐❧í♥❞r✐❝♦ ♦✉ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❝✐❧í♥❞r✐❝♦✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❣❡♥❡r❛❧✐③❛❞♦ ❝♦♠ ✈❛❧♦r❡s ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt U1✱

t❛❧ q✉❡ U0 = Q1/2(U) ❡stá ✧❡♥❝❛✐①❛❞♦✧❡♠ U1 ♣♦✐s ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡

I : U0 → U0 ♥ã♦ é ✉♠ ♦♣❡r❛❞♦r tr❛ç♦✳ ❊♥tã♦ ❞❡✜♥✐♠♦s U1 ❝♦♠♦ ✉♠ ❡s✲

♣❛ç♦ ❞❡ ❍✐❧❜❡rt t❛❧ q✉❡ I :U0 →U1 s❡❥❛ ✉♠ ♦♣❡r❛❞♦r ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t✱ ❡

✉s❛♠♦s ❡st❡ ♦♣❡r❛❞♦r ❝♦♠♦ ♦ ♦♣❡r❛❞♦r J ❞♦ ❚❡♦r❡♠❛ ✶✳✶✸✳

✶✳✺ ■♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛

✶✳✺✳✶ ❉❡✜♥✐çã♦ ❞♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦

❙❡❥❛ W(t) ✉♠ Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❡♠ (Ω,F_,P₎ ❝♦♠ ✈❛❧♦r❡s ♥✉♠ ❡s♣❛ç♦

❞❡ ❍✐❧❜❡rt U✳ ❘❡❝♦r❞❛♠♦s q✉❡ W(t) ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ✭✻✮✳ ❈♦♠

♦ ✐♥t✉✐t♦ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦ ✈❛♠♦s s✉♣♦r q✉❡ λk > 0 ♣❛r❛ q✉❛❧q✉❡r k _∈N✳ ❱❛♠♦s t❛♠❜é♠ ❛ss✉♠✐r q✉❡ ❞❛❞❛ ✉♠❛ ✜❧tr❛çã♦ _{F_t}t≥0 ❡♠ F✱

(i) W(t) é {F_t} ✲ ♠❡♥s✉rá✈❡❧✱

(ii) W(t+h)−W(t) é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ {F_t}✱ _{∀h≥0, ∀t≥0,}

E_{[W(t+h)−W(t)|}F_{t] =}E_{[W(t+h)−W(t)].}

❙❡ ✉♠ Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r W s❛t✐s❢❛③ (i)✱ ❞✐③❡♠♦s q✉❡ W é ❛❞❛♣t❛❞♦

❛ {F_t}✱ s❡ t❛♠❜é♠ ❢♦r s❛t✐s❢❡✐t❛ ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞✐③❡♠♦s q✉❡ _W é ✉♠ _Q ✲ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r ❡♠ r❡❧❛çã♦ ❛ {F_t}✳

◆♦t❛✿ ❉❛❞♦ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ X = {Xt : t ∈ T} ❝♦♠ ✈❛❧♦r❡s ❡♠

♥✉♠ ❡s♣❛ç♦ ♠❡♥s✉rá✈❡❧ (H,Σ)✱ ❛ ✜❧tr❛çã♦ ♥❛t✉r❛❧ é ❞❡✜♥✐❞❛ ❝♦♠♦

F_{t =σ X}−1

s (A) : s ≤t, A∈Σ .

❯♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ é s❡♠♣r❡ ❛❞❛♣t❛❞♦ r❡❧❛t✐✈❛♠❡♥t❡ à s✉❛ ✜❧tr❛çã♦ ♥❛✲ t✉r❛❧✳

❱❛♠♦s ❡♥tã♦ ❞❡✜♥✐r ♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ♣❛r❛ ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s✳ ❋✐✲ ①❡♠♦s T < _∞✳ ❯♠ ♣r♦❝❡ss♦ φ(t)✱ t _∈ [0, T]✱ ❝♦♠ ✈❛❧♦r❡s ♥♦ ❡s♣❛ç♦ ❞♦s

♦♣❡r❛❞♦r❡s ❧✐♠✐t❛❞♦s ❞❡ U → H✱ L = L(U, H)✱ ❞✐③✲s❡ ❡❧❡♠❡♥t❛r s❡ ❡①✐st✐r

✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ t❡♠♣♦s✱ 0 = t0 < t1 < ... < tk = T ❡ ✉♠❛ s✉❝❡ssã♦ φ0, φ1, ..., φk ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❝♦♠ ✈❛❧♦r❡s ❡♠ ▲ t♦♠❛♥❞♦ ✉♠ ♥ú♠❡r♦

✜♥✐t♦ ❞❡ ✈❛❧♦r❡s t❛❧ q✉❡ φm sã♦{F_t

m} ♠❡♥s✉rá✈❡✐s ❡

φ(t) =φm, t ∈(tm, tm+1], m= 0,1, ..., k.

P❛r❛ ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s ♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ é ❞❡✜♥✐❞♦ ♣♦r✿

Z t

0

φ(s)dW(s) =

k−1

X

m=0

φm(Wtm+1∧t −Wtm∧t)

❡ r❡♣r❡s❡♥t❛✲s❡ ♣♦r φ.W(t)✱ t ∈ [0, T]✳ ❱❛♠♦s ❛❣♦r❛ r❡❧❡♠❜r❛r ♦ s✉❜s♣❛ç♦

U0 =Q1/2(U)❞❡U ✐♥tr♦❞✉③✐❞♦ ♥❛ s❡❝çã♦ ❛♥t❡r✐♦r✱ q✉❡ ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

❞❡✜♥✐❞♦ ♣♦r✿

hu, vi0 =

∞

X

k=1

1

λhu, ekihv, eki=hQ

−1/2_{u, Q}−1/2_{vi, u, v∈U} 0

é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

❈♦♥s✐❞❡r❛♠♦s ♦ ❡s♣❛ç♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞tL0

(2) =L(2)(U0, H)

❞❡ U0 ♣❛r❛ H✳ ❘❡❧❡♠❜r❛♠♦s q✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✺✱ ♦ ❡s♣❛ç♦L0(2) t❛♠❜é♠ é

✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛✿

||Ψ_||2_HS0 =

∞

X

h,k=1

|hΨgh, fk_i|2 = ∞

X

h,k=1

λh_|hΨeh, fk_i|2

=||ΨQ1/2||2

HS(U,H)=T r[(ΨQ1/2)(ΨQ1/2)∗],

♦♥❞❡ {gj}n∈N✱ ❝♦♠ gj =

p

λjej✱ {ej}n∈N ❡ {fj}_n∈N sã♦ ❜❛s❡s ♦rt♦♥♦r♠❛❞❛s

❞❡ U0✱U ❡ H✱ r❡s♣❡t✐✈❛♠❡♥t❡✳

❙❡❥❛ Φ(t)✱ t ∈ [0, T]✱ ✉♠ ♣r♦❝❡ss♦ ♠❡♥s✉rá✈❡❧ ❝♦♠ ✈❛❧♦r❡s ❡♠ L0

(2)✱ ❞❡✜✲

♥✐♠♦s ❛s ♥♦r♠❛s✿

|||Φ|||t=

E

Z t

0 ||

Ψ||2

HS0ds

1 2

=

E

Z t

0

T r[(ΨQ1/2)(ΨQ1/2)∗]ds

1 2

, t∈[0, T].

❚❡♦r❡♠❛ ✶✳✶✹ ❙❡ ✉♠ ♣r♦❝❡ss♦ Φ é ❡❧❡♠❡♥t❛r ❡ |||Φ|||T < ∞ ❡♥tã♦ ♦ ♣r♦✲

❝❡ss♦ Φ.W é ✉♠❛ ♠❛rt✐♥❣❛❧❛ ❝♦♥tí♥✉❛✱ ❞❡ q✉❛❞r❛❞♦ ✐♥t❡❣rá✈❡❧ ❝♦♠ ✈❛❧♦r❡s

❡♠ H ❡

E_|Φ.W(t)|2 _=|||Φ|||2

t, 0≤t ≤T. ✭✶✵✮

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r q✉❡ ✭✶✵✮ é ✈❡r✐✜❝❛❞♦ ♣❛r❛ t = tm _≤ T✳

❉❡✜♥❡✲s❡ ζj =W(tj+1)−W(tj), j = 1, ..., m−1✳ ❊♥tã♦

E_|Φ.W(tm)|2 ₌E

m_X−1

j=1

Φ(tj)ζj

2

=E

m_X−1

j=1

|Φ(tj)ζj_|2+2E

m_X−1

i<j=1

hΦ(ti)ζi,Φ(tj)ζj_i.

❱❛♠♦s ❢♦❝❛r✲♥♦s ♥♦ t❡r♠♦ EPm−1

j=1 |Φ(tj)ζj|2✳ ❙❛❜❡♠♦s q✉❡ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛✲

tór✐❛ Φ∗_(t

j)fl é Ftj✲♠❡♥s✉rá✈❡❧✱ ❡ ζj é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ✐♥❞❡♣❡♥❞❡♥t❡

❞❡ F_t

j✳ ❊♥tã♦✱ t❡♠✲s❡

E_|Φ(tj)ζj|2 ₌

m−1

X

l=1

E_{(|hΦ(tj)ζj, fli|}2_{) =}

m_X−1

l=1

E₍E_[|hζj,Φ∗_(tj)fli|2_|F_t

j])

=(tj+1−tj)

m_X−1

l=1

E_(|hQΦ∗_(tj)fl,Φ∗_(tj)fli|)

=(tj+1−tj)

m_X−1

l=1

E_(|Q1/2_Φ∗_(tj)fl|2_{) = (tj+1−tj)||Φ(tj)||}2

HS0.

❯s❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❞❡❞✉③✐♠♦s q✉❡

2E

m−1

X

i<j=1

hΦ(ti)ζi,Φ(tj)ζj_i

=2

m−1

X

i<j=1

E

E _{hΦ(ti)ζi,Φ(tj)ζji|}F_t

i+1

= 0,

♣♦✐s W t❡♠ ✐♥❝r❡♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ♠é❞✐❛ ✵✳

◆♦t❛✿ ❘❡♣❛r❛♠♦s q✉❡ t❛❧ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ é ✉♠❛ ✐s♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ ❞♦s ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛❧ |||._|||T ♣❛r❛ ♦ ❡s♣❛ç♦MT2(H) ❞❛s ♠❛rt✐♥❣❛❧❛s ❝♦♠ ✈❛❧♦r❡s ❡♠

❍✳

❘❡❧❛t✐✈❛♠❡♥t❡ à ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠❛rt✐♥❣❛❧❛✱ ❞❡✜♥✐♥❞♦ s=t′

m < t✱ t❡♠♦s

E

Z t

0

Φ(u)dW(u)_|F_s

=

Z s

0

Φ(u)dW(u) +E

m_X−1

j=m′

Φjζi|Fs

=

Z s

0

Φ(u)dW(u),

✉♠❛ ✈❡③ q✉❡ W t❡♠ ✐♥❝r❡♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ♠é❞✐❛ ✵✳

P❛r❛ ❡st❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❛ ♣r♦❝❡ss♦s ♠❛✐s ❣❡r❛✐s é ❝♦♥✈❡♥✐❡♥t❡ ✐♥t❡r♣r❡t❛r ❡st❡s ♣r♦❝❡ss♦s ❝♦♠♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✱ ❞❡✜♥✐❞❛s ♥♦ ❡s♣❛ç♦ ♣r♦❞✉t♦ ΩT = [0, T]×Ω✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛σ✲á❧❣❡❜r❛ B([0, T])×F✳

❖ ♣r♦❞✉t♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ [0, T] ❡ ❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ P

❞❡♥♦t❛✲s❡ ♣♦r P_T✳

❆❝♦♥t❡❝❡ q✉❡ ❛ σ✲á❧❣❡❜r❛ ❝♦♥s✐❞❡r❛❞❛ ♥ã♦ é ❛❞❡q✉❛❞❛ ❞❡✈✐❞♦ à ♥ã♦ ❛❞❛♣✲

t❛❜✐❧✐❞❛❞❡ ❞♦s ♣r♦❝❡ss♦s ❝♦♥s✐❞❡r❛❞♦s✱ ♣♦rt❛♥t♦ ♥ã♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛✳ ❆ ❡s❝♦❧❤❛ ❝♦rr❡t❛ é ❛ σ✲á❧❣❡❜r❛ P_T ✐♥tr♦❞✉③✐❞❛ ♥❛ ❙❡❝çã♦ ✷✳✸✳✶✳✳ ■r❡♠♦s ❛❣♦r❛ ✈❡r✐✜❝❛r q✉❡ ❛ ❝❧❛ss❡ ❞♦s ✐♥t❡❣r❛♥❞♦s sã♦ ❛♣❧✐❝❛çõ❡s ♠❡♥s✉rá✈❡✐s ❞❡(ΩT,PT)

♣❛r❛ (L0

(2),B(L0(2)))✳

❚❡♦r❡♠❛ ✶✳✶✺ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿

(i) ❙❡ ✉♠❛ ❛♣❧✐❝❛çã♦ Φ : ΩT → L✱ é L✲♣r❡✈✐sí✈❡❧✱ ❡♥tã♦ Φ t❛♠❜é♠ é L0₍₂₎✲

♣r❡✈✐sí✈❡❧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s sã♦ L0

(2)✲♣r❡✈✐sí✈❡✐s✳

(ii) ❙❡Φé ✉♠ ♣r♦❝❡ss♦ L0

(2)✲♣r❡✈✐sí✈❡❧ t❛❧ q✉❡ |||Φ|||T <∞,❡♥tã♦ ❡①✐st❡ ✉♠❛

s✉❝❡ssã♦ {Φn} ❞❡ ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s t❛❧ q✉❡ |||Φ−Φn|||T → 0 q✉❛♥❞♦ n → ∞✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ {en}n∈N✱ {fn}_n∈N ❜❛s❡s ♦rt♦♥♦r♠❛❞❛s ❞❡ U ❡ H r❡s✲

♣❡t✐✈❛♠❡♥t❡✳ ❈♦♠♦ ♦s ♦♣❡r❛❞♦r❡s

fk_⊗ej.u=fk_hej, u_i, u_∈U, k, j_∈N

sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♥s♦s ❡♠ L0

(2) ❡ ♣❛r❛T ∈L0(2) ❛r❜✐trár✐♦✱ hfk⊗ej, TiL0

(2) =λjhT ej, fkiH.

P❡❧❛ ♣r♦♣♦s✐çã♦ ❇ ❞♦ ❆♣ê♥❞✐❝❡✱ ✜❝❛ ❞❡♠♦♥str❛❞♦ (i)✳

❱❛♠♦s ❛❣♦r❛ ❝♦♥s✐❞❡r❛r (ii)✳ ❈♦♠♦ ♦ ❡s♣❛ç♦ L é ❞❡♥s♦ ❡♠ L0

(2) ♣❡❧❛ ♣r♦♣♦✲

s✐çã♦ ❈ ❞♦ ❆♣ê♥❞✐❝❡ ❡①✐st❡ ✉♠❛ s✉❝❡ssã♦ {Φn}n∈N ❞❡ ♣r♦❝❡ss♦s ❡❧❡♠❡♥t❛r❡s

L✲♣r❡✈✐sí✈❡✐s ❡♠ [0, T] t❛❧ q✉❡✿

||Φ(t, ω)−Φn(t, ω)||HS0 ↓ 0,

♣❛r❛ (t, ω) _∈ ΩT✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ |||Φ−Φn|||T ↓ 0✳ ❊♥tã♦ é s✉✜❝✐❡♥t❡

❞❡♠♦♥str❛r q✉❡ ♣❛r❛ A _∈ P_T ❛r❜✐trár✐♦ ❡ ♣❛r❛ q✉❛❧q✉❡r _{ǫ > 0} ❡①✐st❡ ✉♠❛ s♦♠❛ ✜♥✐t❛ Γ ❞❡ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ❞❛ ❢♦r♠❛

(s, t]_×F,0_≤s < t < T, F _∈F_{se{0} ×F, F ∈}F₀ ✭✶✶✮ t❛❧ q✉❡

P_{T{(A\Γ)∪(Γ\A)}< ǫ.} ✭✶✷✮

P❛r❛ ♠♦str❛r ❡st❡ ❢❛❝t♦ ✈❛♠♦s ❞❡♥♦t❛r ♣♦rK ❛ ❢❛♠í❧✐❛ ❞❡ t♦❞❛s ❛s s♦♠❛s ✜✲ ♥✐t❛s ❞❡ ❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛ ✭✶✶✮✱ ❝♦♠s ≤t≤T✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡K é ✉♠

π✲s✐st❡♠❛ ✭✈❡r ❆♣ê♥❞✐❝❡✮✳ ❙❡❥❛G ❛ ❢❛♠í❧✐❛ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s_A∈P_T q✉❡ ♣♦❞❡♠ s❡r ❛♣r♦①✐♠❛❞♦s ♣♦r ❡❧❡♠❡♥t♦s ❞❡ K ✳ ❚❡♠♦s K _⊂G ❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣♦s✐çã♦ ❉ ❞♦ ❆♣ê♥❞✐❝❡ sã♦ ✈❡r✐✜❝❛❞❛s✳ P♦rt❛♥t♦σ(K _{) =}P_{T =}G✳

P♦❞❡♠♦s ❡♥tã♦ ❡st❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❛ t♦❞♦s ♦s ♣r♦✲ ❝❡ss♦s ♣r❡✈✐sí✈❡✐s ❝♦♠ ✈❛❧♦r❡s ❡♠ L0

(2) Φ ❡♠ q✉❡ |||Φ|||T <∞✳

❆té ❛❣♦r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❢♦✐ ❢❡✐t❛ ❝♦♠ ❜❛s❡ ♥❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ ♦ ♦♣❡r❛❞♦r Q é ❞❡ ❝❧❛ss❡ tr❛ç♦✱ só ❛ss✐♠ ✉♠ Q✲♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r

t❡♠ ✈❛❧♦r❡s ❡♠ ❯✳ ❈♦♥t✉❞♦✱ é ♣♦ssí✈❡❧ ❡st❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❞♦ ✐♥t❡❣r❛❧ ❡s✲ t♦❝ást✐❝♦ ❛♦ ❝❛s♦ ❞♦s ♣r♦❝❡ss♦s ❞❡ ❲✐❡♥❡r ❣❡♥❡r❛❧✐③❛❞♦s✱ ♦♥❞❡ ♦ ♦♣❡r❛❞♦r ❞❡ ❝♦✈❛r✐â♥❝✐❛ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡ ❝❧❛ss❡ tr❛ç♦✳ ❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ✐r❡♠♦s ❞❡♥♦t❛r U0 = Q1/2(U) ❝♦♠ ❛ ♥♦r♠❛ ✐♥❞✉③✐❞❛ ||u||0 = ||Q−1/2(u)||✱

u_∈U0✱ ❡ L0(2) =L(2)(U0, H)✳

❈♦♠❡ç❛♠♦s ♣♦r ✐♥tr♦❞✉③✐r ✉♠ t❡♦r❡♠❛ ♥❡❝❡ssár✐♦ ♣❛r❛ ❢❛③❡r ❛ ❣❡♥❡r❛❧✐③❛✲ çã♦ ❞♦ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦ ❛ ♣r♦❝❡ss♦s ♠❛✐s ❣❡r❛✐s✳

❚❡♦r❡♠❛ ✶✳✶✻ ❙❡❥❛ Z ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ✈❛❧♦r❡s ❡♠ U✱ ♠é❞✐❛ ✵✱

❡ ❝♦✈❛r✐â♥❝✐❛ ◗✱ ❡ ❘ ✉♠ ♦♣❡r❛❞♦r ❞❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❞❡ U0 ♣❛r❛ H✳ ❙❡ {Rn_{} ⊂}L0

(2) ❢♦r t❛❧ q✉❡

lim

n→∞||R−Rn||HS0 = 0,

❡①✐st❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ RZ t❛❧ q✉❡

lim

n→∞

E_||RZ−RnZ||2

HS0 = 0.

RZ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ {Rn}

❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞❛ ✐❞❡♥t✐❞❛❞❡

E_|SZ|2 _=||SQ1/2_||2_HS(U,H),

✈á❧✐❞❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s S:U →H✳

❊st❛♠♦s ❡♥tã♦ ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❣❡♥❡r❛❧✐③❛r ❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝♦✳